# What exactly is the relation between the Wiener process and Wiener measure?

The Wiener measure is (in the classical sense) a Gaussian measure on the Banach space $$C[0,1]:=\{f:[0,1] \to \mathbb{R} \mid f\text{ is continuous and } f(0)=1\}$$.

The Wiener process is a stochastic process whose definition can be found in any textbook. In any text, the stochastic integrals or stochastic differential equations use the notation $$dW$$ frequently, which should denote the Wiener measure, I suppose.

However, I cannot figure out the exact relation between the Wiener 'process' and 'measure'. Wikipedia says that the Wiener process induces the Wiener measure but what exactly does that mean?

I am afraid this question might not belong to MO but I ask here. Could anyone please clarify and help me understand?

## 2 Answers

The Wiener measure $$w$$ is the distribution of the Wiener process/random function $$W$$ on $$C[0,1]$$; that is, $$P(W\in A)=w(A)$$ for all Borel sets $$A\subseteq C[0,1]$$. Here "Borel sets" can be replaced by "open sets" or "closed sets".

Equivalently, $$Ef(W)=\int_{C[0,1]}f\,dw$$ for all (say) nonnegative Borel-measurable functions $$f\colon C[0,1]\to\mathbb R$$. Here "nonnegative Borel-measurable" can be replaced by (say) "bounded continuous".

We have to differentiate between the probability measure $$w$$ as a measure on the space $$(C[0,1],\mathcal{B}(C[0,1]))$$, the Wiener measure and the concept of an Ito integral $$\int_{[0,t]} X_s dW_s$$, $$0 \leq t \leq 1$$, where $$W$$ is the usual Wiener process. The Wiener measure is only a special probability measure. Of much more interest is of course the Ito integral. This is definitely not a special case of the integral in the answer of Iosif Pinelis. It requires different concepts. There are many good introductions to the Ito integral, f.i. Karatzas/Shreve (1988), Brownian Motion and Stochastic Calculus, to mention only one.